Using quantum corrections from massless fields conformally coupled to gravity, we study the possibility ofavoiding singularities that appear in the flat Friedmann–Robertson–Walker model. We assume that theuniverse contains a barotropic perfect fluid with the state equation p = ωρ, where p is the pressure andρ is the energy density. We study the dynamics of the model for all values of the parameter ω and alsofor all values of the conformal anomaly coefficients α and β. We show that singularities can be avoidedonly in the case where α > 0 and β < 0. To obtain an expanding Friedmann universe at late timeswith ω > −1 (only a one-parameter family of solutions, but no a general solution, has this behavior atlate times), the initial conditions of the nonsingular solutions at early times must be chosen very exactly.These nonsingular solutions consist of a general solution (a two-parameter family) exiting the contractingde Sitter phase and a one-parameter family exiting the contracting Friedmann phase. On the other hand,for ω < −1 (a phantom field), the problem of avoiding singularities is more involved because if we consideran expanding Friedmann phase at early times, then in addition to fine-tuning the initial conditions, wemust also fine-tune the parameters α and β to obtain a behavior without future singularities: only a oneparameterfamily of solutions follows a contracting Friedmann phase at late times, and only a particularsolution behaves like a contracting de Sitter universe. The other solutions have future singularities.